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A289786
p-INVERT of the odd positive integers (A005408), where p(S) = 1 - S - S^2.
3
1, 5, 20, 77, 291, 1098, 4149, 15689, 59332, 224369, 848447, 3208370, 12132345, 45878109, 173486772, 656035301, 2480778763, 9380993978, 35473960589, 134143768193, 507260826084, 1918192318185, 7253589435975, 27429241169378, 103722891648049, 392225150722037
OFFSET
0,2
COMMENTS
Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the INVERT transform of s, so that p-INVERT is a generalization of the INVERT transform (e.g., A033453).
See A289780 for a guide to related sequences.
FORMULA
G.f.: (-1 - x^2 - 2 x^3)/(-1 + 5 x - 6 x^2 + 5 x^3 + x^4).
a(n) = 5*a(n-1) - 6*a(n-2) + 5*a(n-3) + a(n-4).
MATHEMATICA
z = 60; s = x (1 + x)/(1 - x)^2; p = 1 - s - s^2;
Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A005408 *)
Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1] (* A289786 *)
LinearRecurrence[{5, -6, 5, 1}, {1, 5, 20, 77}, 30] (* Harvey P. Dale, May 06 2018 *)
CROSSREFS
Sequence in context: A269708 A295347 A270985 * A129869 A271887 A079737
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Aug 10 2017
STATUS
approved